Interacting bosons in a lattice

Bose Hubbard Model

\[\begin{aligned} \hat{H}=& \int d^{3} \mathbf{r} \cdot \psi^{\dagger}(\mathbf{r})\left[-\frac{\hbar^{2} \nabla^{2}}{2 m}+V_{\text {ext. }}(\mathbf{r})\right] \psi(\mathbf{r}) +\frac{1}{2} \int \psi^{\dagger}(\mathbf{r}) \psi^{\dagger}\left(\mathbf{r}^{\prime}\right) \underbrace{V\left(\mathbf{r}^{\prime}-\mathbf{r}\right)}_{V_c + V_d} \psi\left(\mathbf{r}^{\prime}\right) \psi(\mathbf{r}) \cdot d^{3} \mathbf{r} d^{3} \mathbf{r}^{\prime} \end{aligned}\]

\(\hspace{9.5cm}\)nearest-neighbour \(\hspace{1cm}\Bigg\Downarrow \hspace{1cm}\psi(r) = \sum_i w_{R_i}(r) \cdot a_i\)

\[\boxed{H_{BH} = -t\cdot \sum_{\langle i,j \rangle} a_i^{\dagger}a_j + \frac{U}{2} \sum_i n_i(n_i -1) + V \sum_{\langle i, j \rangle} n_i n_j}\]
\[\small t_{i, j} = \int dr \cdot w^*_{R_i}(r) \cdot \hat{H} \cdot w_{R_j}(r)\]

\[\small V_c(r) = \frac{4π\hbar^2 a_s}{m}\cdot\delta(r) = g\cdot\delta(r) U_i \hspace{1cm}\longrightarrow \hspace{1cm} U_i = g \int dr \cdot |w_{R_i}(r)|^4\]

\[\small V_d(r, r') = \frac{\mu_0 \mu_m^2}{4\pi} \cdot \frac{(1 - 3\cos^2\theta)}{(r - r')^3} \hspace{1cm}\longrightarrow \hspace{1cm} V_{i, j} = \int dr^3 dr'^3 \cdot|w_{R_i}(r')|^2 \cdot V_d(r - r') \cdot |w_{R_j}(r)|^2 \]

Bose Hubbard Model

Phases of the BHM

Mott Insulator

Superfluid

\[\small H = -t\sum_{\langle i, j \rangle} a_i^{\dagger}a_j\hspace{0.5cm}\longrightarrow\hspace{0.5cm}|\Psi_{SF}\rangle= \frac{1}{N!} (\sum_{i=1}^M a_i^{\dagger})^N |{0}\rangle \hspace{0.5cm}\]

Exact Diagonalization

Consider N bosons in M lattice sites.

• \[\text{Enumerate basis set: }\hspace{0.5cm} |n_1, n_2, ..., n_M\rangle \hspace{0.5cm} s.t. \hspace{0.5cm} \sum_{i=1}^M n_i = N\]
  
• \[\text{Construct hamiltonian matrix: }\hspace{0.5cm} \langle n_1, n_2, ..., n_M| \hspace{0.1cm}\hat{H} \hspace{0.1cm}| n_1', n_2', ..., n_M'\rangle\]
  
• \[\text{Diagonalize the hamiltonian: } \hspace{0.5cm} H|\psi\rangle = E|\psi\rangle\]

Results (ED)

Dimensionality scaling

Mean Field Theory

\[\small \underbrace{H = -t\sum_{\langle i, j \rangle} a_i^{\dagger}a_j + \frac{U}{2}\sum_i n_i(n_i - 1)}_{\text{coupled lattice sites}} \hspace{0.5cm}\longrightarrow\hspace{0.5cm} \underbrace{H \{\Psi \} = \sum_i-zt \cdot (\Psi^*a_i + \Psi a_i^{\dagger} - |\Psi|^2) + \frac{U}{2}n_i(n_i -1)}_{\text{de-coupled lattice sites}}\]

Results (MFT)

Results (MFT, contd.)

Cluster Mean Field Theory

\[\small \underbrace{H = -t\sum_{\langle i, j \rangle} a_i^{\dagger}a_j + \frac{U}{2}\sum_i n_i(n_i - 1)}_{\text{coupled lattice sites}} \hspace{0.5cm}\longrightarrow\hspace{0.5cm} \underbrace{H \{\Psi_i \} = \sum_C H_{exact} + \sum_{C, C'}H_{MFT}\{ \Psi_i \}}_{\text{de-coupled clusters of sites}}\]

Results (CMFT)

Dipolar bosons in a lattice

Results (MFT)

Results (MFT, contd.)

Path Integral QMC

Evaluate the partition function \(Z = Tr(e^{\beta\hat{H}})\).

\[\begin{align} Z &= \sum_{|n_1\rangle} \langle n_1 |\left (e^{\frac{\beta}{M}\hat{H}} \right )^M |n_1 \rangle \\ &= \lim_{M\to \infty} \ \ \sum_{|n_1\rangle} \langle n_1 |\left (1 + \frac{\beta}{M}\hat{H} \right)^M |n_1 \rangle \\ &\approx \sum_{\{|n_i\rangle\}} \langle n_1 |\left (1 + i\Delta t\hat{H} \right) |n_2 \rangle \cdot \langle n_2 |\left (1 + i\Delta t\hat{H} \right) |n_3 \rangle \dots \langle n_M |\left (1 + i\Delta t\hat{H} \right) |n_1 \rangle \\ &\approx \sum_{C_i \in C} w(C_i) \end{align}\]

• Map d-dimensional quantum system to (d+1) classical system (\(\beta \rightarrow iM\Delta t\)).

• Formulated as a classical system determined by a configuration - string of M basis elements; \(C \equiv |n_1 \rangle \rightarrow |n_2 \rangle \rightarrow \dots \rightarrow |n_M \rangle\rightarrow |n_1 \rangle\).

Worldline representation